pi-base · felixpernegger · Dec 27, 2025 · Dec 27, 2025 · Dec 28, 2025 · Dec 28, 2025
diff --git a/properties/P000087.md b/properties/P000087.md
@@ -17,3 +17,4 @@ Contrary to Munkres or Willard, we do not assume any separation axiom like {P3},
 #### Meta-properties
 
 - This property is preserved by arbitrary products.
+- This property is preserved by $\Sigma$-products.
diff --git a/properties/P000186.md b/properties/P000186.md
@@ -13,3 +13,4 @@ Homeomorphic to a subspace of a space that is {P87} and {P187}.
 
 - This property is hereditary.
 - This property is preserved by countable products.
+- This property is preserved by $\Sigma$-products.
diff --git a/spaces/S000035/properties/P000062.md b/spaces/S000035/properties/P000062.md
@@ -0,0 +1,7 @@
+---
+space: S000035
+property: P000062
+value: false
+---
+
+Consider the open cover $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $\mathcal{U}\subseteq \mathcal{O}$ such that $\bigcup \mathcal{U}$ dense. But then $\bigcup\mathcal{U}=X$, since otherwise we would find an $a \in X$ with $[a,\omega_1)\subseteq X \setminus \bigcup\mathcal{U}$. Therefore $X$ would be a countable union of countable sets and thus countable.
-Consider the open cover $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $\mathcal{U}\subseteq \mathcal{O}$ such that $\bigcup \mathcal{U}$ dense. But then $\bigcup\mathcal{U}=X$, since otherwise we would find an $a \in X$ with $[a,\omega_1)\subseteq X \setminus \bigcup\mathcal{U}$. Therefore $X$ would be a countable union of countable sets and thus countable.
+Consider the open cover $\mathcal{O}=\{[0,\alpha)\mid \alpha \in X\}$.
+Every countable subset of $\omega_1$ has an upper bound in $\omega_1$.
+So if $\mathcal U$ is a countable subcollection of $\mathcal O$, there is some $\alpha\in\omega_1$ such that
+$\bigcup\mathcal U\subseteq[0,\alpha]$, which is not dense in $X$.
-Consider the open cover $\mathcal{O}=\{[0,a)\mid a \in X\}$. If $X$ were weakly Lindelöf, we would find a countable $\mathcal{U}\subseteq \mathcal{O}$ such that $\bigcup \mathcal{U}$ dense. But then $\bigcup\mathcal{U}=X$, since otherwise we would find an $a \in X$ with $[a,\omega_1)\subseteq X \setminus \bigcup\mathcal{U}$. Therefore $X$ would be a countable union of countable sets and thus countable.
+Consider the open cover $\mathcal{O}=\{[0,\alpha)\mid \alpha \in X\}$.
+Every countable subset of $\omega_1$ has an upper bound in $\omega_1$.
+So if $\mathcal U$ is a countable subcollection of $\mathcal O$, there is some $\alpha\in\omega_1$ such that
+$\bigcup\mathcal U\subseteq[0,\alpha]$, which is not dense in $X$.
diff --git a/spaces/S000035/properties/P000082.md b/spaces/S000035/properties/P000082.md
@@ -0,0 +1,7 @@
+---
+space: S000035
+property: P000082
+value: true
+---
+
+For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).
-For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).
+For $alpha \in X$ consider the open neighborhood $U_\alpha := [0,\alpha+1)\subseteq X$. By the definition of $X$, $U_\alpha$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).
-For $a \in X$ consider the open neighborhood $U_a := [0,a+1)\subseteq X$. By the definition of $X$, $U_a$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).
+For $alpha \in X$ consider the open neighborhood $U_\alpha := [0,\alpha+1)\subseteq X$. By the definition of $X$, $U_\alpha$ is {P57} {P190} and thus {P53} [(Explore)](https://topology.pi-base.org/spaces?q=Countable+%2B+Ordinal+space+%2B+%7EMetrizable).
Original file line number	Diff line number	Diff line change
Expand Up		@@ -17,3 +17,4 @@ Contrary to Munkres or Willard, we do not assume any separation axiom like {P3},
		#### Meta-properties

		- This property is preserved by arbitrary products.
		- This property is preserved by $\Sigma$-products.
Original file line number	Diff line number	Diff line change
Expand Up		@@ -13,3 +13,4 @@ Homeomorphic to a subspace of a space that is {P87} and {P187}.

		- This property is hereditary.
		- This property is preserved by countable products.
		- This property is preserved by $\Sigma$-products.